integral_integral_symm — Mathlib

English

If L is a continuous linear map on E×F to G and f,g are integrable as appropriate, then the integral of L composed with (f,g) equals the sum of integrals of L composed with f and with g.

Русский

Пусть L — непрерывное линейное отображение E×F→G и f,g интегрируемы; тогда интеграл от L∘(f,g) равен суммы интегралов от L∘f и L∘g.

LaTeX

$$$$\\int_{μ×ν} L(f(x), g(y)) = \\int_{μ} L(f) + \\int_{ν} L(g).$$$$

Lean4

/-- Reversed version of **Fubini's Theorem** (symmetric version). -/
theorem integral_integral_symm {f : α → β → E} (hf : Integrable (uncurry f) (μ.prod ν)) :
    ∫ x, ∫ y, f x y ∂ν ∂μ = ∫ z, f z.2 z.1 ∂ν.prod μ :=
  (integral_prod_symm _ hf.swap).symm

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